676 MODELS IN HYDRAULIC ENGINEERING
water and Saint Venant equations (see Section 8.2.2) involving variation of
velocity and/or concentration in space and time.
The simplest partial differential equations (PDEs) frequently occur-
ring in engineering are linear second order equations which in fluid
dynamics often take the form of hyperbolic (wave), parabolic (diffusion)
or elliptical (Laplace) equations. For the solution of PDEs initial and /or
boundary conditions must be given.
The solution of PDEs requires in most cases recourse to numerical
solutions based on finite difference, finite element or finite volume
methods with explicit or implicit solutions; numerical stability, conver-
gence, dissipation and dispersion are important aspects of numerical
models. The method of characteristics transforms the PDEs into ordinary
differential equations along certain characteristics resulting in simplified
step solutions.
There are many computer packages available for the solution of
hydraulic engineering problems, but their successful use requires a good
background knowledge of underlying hydraulic physical and mathematical
principles. Many of these packages relate to specific areas, e.g. Mike 11
(open channel flow), Flowmaster (pipelines and pressure surges)
DAMBRK (dam-break) (see also Section 7.5.2), etc.
For a general discussion of the mathematical background of PDEs
and their solution see, e.g. Jeffrey (2003); for an overview of the develop-
ment of software (and hardware) in computational hydraulics see, e.g.
Abbott (1991), Anderson (1995) and Abbott and Minns (1997); for the
background to turbulence modelling see, e.g. ASCE (1988) and Rodi
(1996). Zinkiewicz and Taylor (2000) provide a detailed account of the
finite element method, Vreugdenhill (1994) gives details of numerical
modelling of shallow water flows and Chadwick et al. (2004) give examples
of applications of computational hydraulics.
Computational fluid dynamics (CFD) has an important role to play
in hydraulic engineering design – see, e.g. Verwey (1983). There have been
rapid developments e.g. in the application of numerical techniques to
overfall spillway design – see Spaliviero and Seed (1998), Song and Zhou
(1999) and Assy (2001). The use of computational hydraulics is particu-
larly widespread in river engineering – see, e.g. Cunge, Holly and Verwey
(1980) – including computation of the flow field round groynes and bridge
abutments (Biglari and Sturm, 1998). Bürgisser (1999) gives a general
overview of the solution of the free water surface at hydraulic structures
using numerical modelling (including an extensive bibliography of the
subject). The application of numerical techniques in the computation of
pressure transients is covered e.g. in Wilie and Streeter (1993). For a brief
discussion of numerical modelling in coastal engineering see Chapter 15.
